We correct an earlier theorem and reprove its consequences regarding \(c\)-BRDs with \(v \equiv 5, 8 \pmod{12}\). The original conclusions remain valid.

The type of a vertex \(v\) in a \(p\)-page book-embedding is the \(p \times 2\) matrix of nonnegative integers

\[{r}(v) =
\left(
\begin{array}{ccccc}
l_{v,1} & r_{v,1} \\
. & . \\
. & . \\
. & . \\
l_{v,p} & r_{v,p} \\
\end{array}
\right),\]

where \(l_{v,i}\) (respectively, \(r_{v,i}\)) is the number of edges incident to \(v\) that connect on page \(i\) to vertices lying to the left (respectively, to the right) of \(v\). The type number of a graph \(G\), \(T(G)\), is the minimum number of different types among all the book-embeddings of \(G\). In this paper, we disprove the conjecture by J. Buss et al. which says for \(n \geq 4\), \(T(L_n)\) is not less than \(5\) and prove that \(T(L_n) = 4\) for \(n \geq 3\).

Let \(T\) be a chemical tree, i.e. a tree with all vertices of degree less than or equal to \(4\). We find relations for the \(0\)-connectivity and \(1\)-connectivity indices \({}^0\chi(T)\) and \({}^1\chi(T)\), respectively, in terms of the vertices and edges of \(T\). A comparison of these relations with the coefficients of the characteristic polynomial of \(T\) associated to its adjacency matrix is established.

Given a regular action of a finite group \(G\) on a set \(V\), we consider the problem of the existence of an incidence structure \(\mathcal{I} = (V, \mathcal{B})\) on the set \(V\) whose full automorphism group \(Aut(\mathcal{I})\) is the group \(G\) in its regular action. Using results on graphical and digraphical regular representations \(([2,7], [1])\), we show the existence of such an incidence structure for all but four small finite groups.

For a finite field \({F} = {F}(q)\), where \(q = p^n\) is a prime power, we will introduce the notion of equivalence of subsets of \(F\) which stems out of the equivalence of cyclic difference sets, and give the formulae for the number of equivalence classes of \(k\)-subsets of \(F\) as well as for the number of equivalence classes of subsets of \(F\) by using Pólya’s theorem of counting.

We present an algorithmic construction of anti-Pasch Steiner triple systems for orders congruent to \(9\) mod \(12\). This is a Bose-type method derived from a particular type of \(3\)-triangulations generated from non-sum-one-difference-zero sequences (\(NS1D0\) sequences). We introduce \(NS1D0\) sequences and describe their basic properties; in particular, we develop an equivalence between the problem of finding \(NS1D0\) sequences and a variant of the \(n\)-queens problem. This equivalence, and an algebraic characterization of the \(NS1D0\) sequences that produce anti-Pasch Steiner triple systems, form the basis of our algorithm.

For vertices \(u\) and \(v\) in a nontrivial connected graph \(G\), the closed interval \([u,v]\) consists of \(u\), \(v\), and all vertices lying in some \(u-v\) geodesic of \(G\). For \(S \subseteq V(G)\), the set \(I[S]\) is the union of all sets \(I[u,v]\) for \(u,v \in S\). A set \(S\) of vertices of a graph \(G\) is a geodetic set in \(G\) if \(I[S] = V(G)\). The minimum cardinality of a geodetic set in \(G\) is its geodetic number \(g(G)\). A subset \(T\) of a minimum geodetic set \(S\) in a graph \(G\) is a forcing subset for \(S\) if \(S\) is the unique minimum geodetic set containing \(T\). The forcing geodetic number \(f(S)\) of \(S\) in \(G\) is the minimum cardinality of a forcing subset for \(S\), and the upper forcing geodetic number \(f^+(G)\) of the graph \(G\) is the maximum forcing geodetic number among all minimum geodetic sets of \(G\). Thus \(0 \leq f^+(G) \leq g(G)\) for every graph \(G\). The upper forcing geodetic numbers of several classes of graphs are determined. It is shown that for every pair \(a,b\) of integers with \(0 \leq a \leq b\) and \(b \geq 1\), there exists a connected graph \(G\) with \(f^+(G) = a\) and \(g(G) = b\) if and only if \((a, b) \notin \{(1, 1), (2,2)\}\).

We give necessary and sufficient conditions for the existence of a decomposition of the complete graph into stars which admits either a cyclic or a rotational automorphism.

This paper deals with combinatorial aspects of designs for two-way elimination of heterogeneity for making all possible paired comparisons of treatments belonging to two disjoint sets of treatments. Balanced bipartite row-column (BBPRC) designs have been defined which estimate all the elementary contrasts involving two treatments one from each of the two disjoint sets with the same variance. General efficiency balanced row-column designs (GEBRC) are also defined. Some general methods of construction of BBPRC designs have been given using the techniques of reinforcement, deletion (addition) of column or row structures, merging of treatments, balanced bipartite block (BBPB) designs, juxtaposition, etc. Some methods of construction give GEBRC designs also.